BCJR Algorithm. Veterbi Algorithm (revisted) Consider covolutional encoder with. And information sequences of length h = 5

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1 Chapter 2 BCJR Algorithm Ammar Abh-Hhdrohss Islamic University -Gaza ١ Veterbi Algorithm (revisted) Consider covolutional encoder with And information sequences of length h = 5 The trellis diagram has h + m + 1 timeslots which equals 8 in our case Consider received sequence as Slide ٢ ١

2 Slide ٣ From Fig on the text book, we can see that Slide ٤ ٢

3 SOVA The Soft-Output Viterbi Algorithm (SOVA) was first introduced in We describe SOVA for convolutional code with R = 1/n on binary input, AWGN channel. We assume that priori probabilities are not equally likely p(u L ) and L = 0,., h-1. Slide ٥ Log-likelihood metric Let us define the log-likelihood ratio or the L-value of a received symbol r at the output of channel with binary inputs v = 1 Similarly the L-value of an information bit u is defined as Using Bay s rule if v is equally likely Slide ٦ ٣

4 Log-likelihood metric A large positive value of L(r) indicates a high reliability that v = +1. A large negative value of L(r) indicates a high reliability that v = -1. A close to zero value of L(r) indicates a decision a bout the value of v based only on r is unreliable. The same a large positive value of L(u) indicates a high reliability that u = +1 Slide ٧ Log-likelihood metric It can be shown that the L value is equal to (left as exercise for the students) Where is defined as channel reliability factor Slide ٨ ٤

5 BCJR algorithm P w P vˆ v r Veterbi Algorithm minimizes the WER E that is So it is minimizes the error probability between the transmitted and received codeword. In BCJR algorithm, we are interested in minimizing the bit error probability. This is done by maximizing the posteriori probability That is why BCJR decoder is also called Maximum Posteriori Probability decoder (MAP) Slide ٩ BCJR algorithm We don t assume that the information bits are equally likely. The algorithms calculates the a posteriori L-values Called APP L-values of each information bit, the decoder output is given by Slide ١٠ ٥

6 We start our development of the BCJR algorithm by rewriting the APP value as Where U L+ is the set of all information sequences u such as u l = 1, v is the transmitted codeword corresponding to the information sequence u. So we can rewrite the expression o f the APP L values as Where U L- is the set of all information sequences u such as u l = - 1 Slide ١١ The L values can be calculated using the previous formula but still it suffers from high degree of complexity. We can rewrite the a posteriori probability as Where l+ is the set of all state pairs s l = s and s l+1 = s that corresponds to the input bit u l = + 1. Reforming P(u l = -1/r) in the same way and sub in the L value Slide ١٢ ٦

7 Where l- is the set of all state pairs s l = s and s l +1 = s that corresponds to the input bit u l = - 1. The joint pdf p(s,s,r) can be found recursively, starting from Where r t <l represents the portion of the received r before the time l and Where r t > l represents the portion of the received r after the time l. Now application of Bay s rule Slide ١٣ Defining So the joint pdf can be rewritten as We can write expression for α l+1 (s) as Where σ l is the set of all states at time l. Slide ١٤ ٧

8 l (s ) can be written as Where σ l+1 is the set of all states at time l+1. The forward recursion starts from And the backward recursion starts from Slide ١٥ We can write the branch metric as Which yields We can drop the constant to achieve Slide ١٦ ٨

9 The priori probability can be written as Slide ١٧ The L value depend of the value of u, thus Again if we drop the constants Slide ١٨ ٩

10 Using the log-domain enable using And the log-domain metrics are Slide ١٩ Writing the expression for the pdf p(s, s, r) and the APP L- value L(u l ) as: Slide ٢٠ ١٠

11 Using the following math expression We can formula the L value as Slide ٢١ Steps of Log-Domain BCJR algorithm Step1: calculate the forward and backward metrics using Step 2 Compute the branch metric using Slide ٢٢ ١١

12 Steps of Log-Domain BCJR algorithm Step3: calculate the forward metrics using Step 4 Compute the backward metric using Step 5 compute the APP-L values using Slide ٢٣ Steps of Log-Domain BCJR algorithm Step6: (Optional) compute the hard decisions using Slide ٢٤ ١٢

13 Example We will consider the BCJR decoding of a (2, 1, 1) systematic Recursive Convolutional code on AWGN with generator matrix Slide ٢٥ Let u = (u0, u1, u2, u3) denote the input vector of length 4 and v = (v0, v1, v2, v3) denotes the codeword of length 8. We assume Es/ N0 = ¼(-6.02) db The received vector r = (+0.8, +0.1; +1.0, -0.5; -1.8, +1.1; 1.6, -1.6) Slide ٢٦ ١٣

14 The rate of the terminated code is R = h/n =3/8. Eb/ N0 = Es/RN0 = 2/3 Assuming that the information bits are equaly likely La(ul) = 0 Lc = 4 Eb/N0 = 1 Slide ٢٧ Slide ٢٨ ١٤

15 Slide ٢٩ Now compute the log-domain forward metrics Slide ٣٠ ١٥

16 Similarly compute the log-domain backward metrics Slide ٣١ Finally we compute the app L -values Slide ٣٢ ١٦

17 Slide ٣٣ Example Slide ٣٤ ١٧

18 Finding the L-values Slide ٣٥ Then he hard decision outputs of the Max-log-MAP decoder is u = (-1, +1, -1) Slide ٣٦ ١٨

Lattice Coding and its Applications in Communications

Lattice Coding and its Applications in Communications Alister Burr University of York alister.burr@york.ac.uk Introduction to lattices Definition; Sphere packings; Basis vectors; Matrix description Codes